In reply to Marq:
These are near the downtown or have good public transport connections. Suprisingly, I hardly found English versions, but all linked sites show the address, the open hours (nyitvatartas) somewhere. Hetfo is Monday, and so on. Many have FB profile as well.
Only boulder:
Mountex boulder club:
http://boulder.mountex.hu/ (Downtown, near subway.)
Monkey Boulder club:
http://monkeyboulder.hu/ (Largest, but in the outskirt)
Ujjerõ :
https://www.google.hu/search?q=ujjer%C5%91+m%C3%A1sz%C3%B3terem&ie=utf-...
with rope too:
Spider Club:
http://www.spiderclub.hu/en/The-Hall/61/0 (Most of the routes are more or less overhanging.)
Big Wall:
http://www.bigwall.hu/nyitvatartas/
Mászóbirodalom:
http://maszobirodalom.hu/en/opening-hours (Worst equipments, but I was 2 years ago)
BME Sportkozpont:
http://bmefitness.hu/szolgaltatasok?falmaszas#gallery-details-26 (In the complex of the technical university, but the wall is little)